In a bowl, add beef broth, Worcestershire sauce, Dijon mustard, cooking sherry, garlic, Herbes de Provence and smoked paprika. Whisk until well combined.
Place roast beef slices in a large re-sealable plastic bag and pour on the marinade. Close the bag and shake it around to coat the beef.
Place the bag in a pie plate and chill for 2 hours.
A couple hours later, slice the buns like a hot dog (not completely) and place them on a baking sheet. Transfer under the broiler for 2 minutes.
Remove the buns from the heat and let them cool off.
Take the marinated beef out of the fridge and let it get back to room temperature.
In a large skillet over medium heat, add olive oil and butter.
When hot, add onions and 1 tsp. ground sea salt. Stir well, reduce the heat to medium-low and cook slowly for 10 to 12 minutes or until caramelized and soft, stirring often.
Stir in the peppers and sauté for 4 minutes.
Add the mushrooms and season generously with freshly ground black pepper; stir well.
Increase the heat to medium and cover with the lid; cook the veggies for 4 minutes. Halfway through cooking, give another good stir.
Using a slotted spoon, transfer the veggies to a bowl and set aside.
Add the beef including the marinade and evenly spread the slices out. Cover and warm up the meat, about 2 to 3 minutes.
To assemble the sandwiches, place slices of beef to cover the bottom portion of the bread. Top with a nice layer of vegetable mixture and add a couple slices of cheese per sandwich.
Place them on a baking sheet open-face and transfer the buns under the broiler until the cheese is melted, about 3 minutes. Serve immediately.